On the existence of solutions and one-step method for functional differential equations with parameters. (English) Zbl 0817.34006

Der Gegenstand der Arbeit ist die Lösung der Randwertaufgabe für die Volterrasche neutrale Funktional-Differentialgleichung der Form \[ x'(t)= f(t,x(.), x'(.),\lambda),\quad t\in I,\tag{1} \]
\[ x(a)= x_ p,\;L(x(.), \lambda)= \theta\in \mathbb{R}^ q,\tag{2} \] wobei \(f: I\times C^ 1(I, \mathbb{R}^ p)\times C(I,\mathbb{R}^ p)\times \mathbb{R}^ q\to \mathbb{R}^ p\), \(L: C^ 1(I, \mathbb{R}^ p)\times \mathbb{R}^ q\to \mathbb{R}^ q\), \(x_ p\in \mathbb{R}^ p\) gegeben sind und \(I= [a,b]\), \(a< b\). Die Aufgabe ist durch die Substitution \(y= x'\) äquivalent mit einer nichtlinearen Integralgleichung. Diese neue Aufgabe löst der Autor mittels Verwendung des Einschrittverfahrens für \(y\) kombiniert mit der Newtonschen Methode für \(\lambda\). Im weiteren wird die Konvergenz der erreichten Lösungsfolgen untersucht. Die erreichten Ergebnisse sind eine Erweiterung der Ergebnisse früherer Arbeiten des Autors [Math. Nachr. 71, 237-247 (1976; Zbl 0384.34046); Math. Nachr. 125, 7-28 (1986; Zbl 0587.34049); Computing 43, 343-359 (1990; Zbl 0689.65053)].


65J99 Numerical analysis in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65Q05 Numerical methods for functional equations (MSC2000)
35K05 Heat equation
Full Text: EuDML


[1] C. W. Cryer, L. Tavernini: The numerical solution of Volterra functional differential equations by Euler’s method. SIAM J. Numer. Anal. 9 (1972), 105-129. · Zbl 0244.65085
[2] A. Feldstein, R. Goodman: Numerical solution of ordinary and retarded differential equations with discontinuous derivatives. Numer. Math. 21 (1973), 1-13. · Zbl 0266.65056
[3] P. Henrici: Discrete Variable Methods in Ordinary Differential Equations. New York, 1962. · Zbl 0112.34901
[4] Z. Jackiewicz: One-step methods for the numerical solution of Volterra functional differential equations of neutral type. Applicable Anal. 12 (1981), 1-11. · Zbl 0469.65060
[5] Z. Jackiewicz: One-step methods of any order for neutral functional differential equations. SIAM J. Numer. Anal. 21 (1984), 486-511. · Zbl 0562.65056
[6] T. Jankowski, M. Kwapisz: On the existence and uniqueness of solutions of boundary-value problem for differential equations with parameter. Math. Nachr. 7i (1976), 237-247. · Zbl 0384.34046
[7] T. Jankowski: Boundary value problems with a parameter of differential equations with deviated arguments. Math. Nachr. 125 (1986), 7-28. · Zbl 0587.34049
[8] T. Jankowski: One-step methods for retarded differential equations with parameters. Computing A3 (1990), 343-359. · Zbl 0689.65053
[9] T. Jankowski: Convergence of multistep methods for retarded differential equations with parameters. Applicable Anal. 57(1990), 227-251. · Zbl 0716.65065
[10] T. Pomentale: A constructive theorem of existence and uniqueness for the problem \(V = f(x, y, \lambda)\), \(y(a) = \alpha\), \(y(b) = \beta\).\(. ZAMM 56 (1976), 387-388.\) · Zbl 0338.34019
[11] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. New York, Heidelberg, Berlin, 1980. · Zbl 0553.65004
[12] L. Tavernini: One-step methods for the numerical solutions of Volterra functional differential equations. SIAM J. Numer. Anal. 8 (1971), 786-795. · Zbl 0231.65070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.